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Inscription and p-adic twistors

SEMINARS, SEMINARS: NUMBER THEORY

When: February 5, 2025
3:00 pm - 4:00 pm
Where: Science Center 507
Address: 1 Oxford Street, Cambridge, MA 02138, United States
Speaker: Sean Howe (University of Utah)

Inspired by a construction of Simpson for irreducible local systems over compact Kahler manifolds, both Fargues and Liu-Zhu have conjectured that p-adic local systems on smooth rigid analytic varieties over p-adic fields should admit associated p-adic twistor bundles. We formulate and prove a version of this conjecture using the theory of inscribed v-sheaves, which is a simple differential extension of Scholze’s approach to p-adic geometry by replacing a classical object with its functor-of-points on perfectoid spaces. As an application, we explain how to obtain a non-trivial inscribed structure on p-adic Lie torsors over smooth rigid analytic varieties that allows us, in particular, to compute Banach-Colmez Tangent Bundles and differentiate Hodge-Tate period maps and their lattice refinements. In the case of infinite level local and global Shimura varieties this agrees with a natural inscribed structure constructed by extending a moduli interpretation to the inscribed setting.

A pretalk will be giving at 2:00 in room 530.

Pretalk: Modern p-adic geometry

Abstract: The basic building blocks of p-adic geometry have shifted in the past fifteen years from the Noetherian convergent power series rings of Tate’s theory of rigid analytic spaces, which mirrors the classical theory of complex analytic spaces, to the more exotic perfectoid rings that provide the test objects in Scholze’s theory of diamonds and v-sheaves and are characterized by the existence of approximate p-power roots. We will give some simple examples contrasting the behaviors of these types of rings and discuss some of the reasons for this shift in perspectives.